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Statement

Definition 1 (p. 2). For a point $B$ on the boundary of a set, the diameter of $B$ is the supremum of the distances from $B$ to points of the set. The set, or its boundary curve, has constant diameter when all boundary points have the same diameter.

The paper remarks (p. 2) that such sets are extremal: adding a small area at the boundary increases the diameter, and the area is locally maximal, the disc having the largest area. It takes this as the reason to use them as tortoises before cutting. Choosing $q(\varphi)$ piecewise constant in the construction of sets of constant width (Kawohl and Sweers) gives sets of constant diameter (p. 2), and the family $D_\epsilon$ of equation (4.5) is built this way with constant diameter $2$.

Source. Helmut Ruhland, No new lower bound for the density of planar sets avoiding unit distances, arXiv:2408.10076, read in the v4 named on the source card, p. 2.

Read depth. Claims checked: the definition was read clause by clause on the printed page. The extremality remark is stated without proof.

Bears on

• Problem 1070: only through the construction of equation (4.5), which, by the paper's own correction, gives no new lower bound on $m_1$.